Surface photogalvanic effect in Ag2Te

The bulk photovoltaic effect (BPVE) in non-centrosymmetric materials has attracted significant attention in recent years due to its potential to surpass the Shockley-Queisser limit. Although these materials are strictly constrained by symmetry, progress has been made in artificially reducing symmetry to stimulate BPVE in wider systems. However, the complexity of these techniques has hindered their practical implementation. In this study, we demonstrate a large intrinsic photocurrent response in centrosymmetric topological insulator Ag2Te, attributed to the surface photogalvanic effect (SPGE), which is induced by symmetry reduction of the surface. Through diverse spatially-resolved measurements on specially designed devices, we directly observe that SPGE in Ag2Te arises from the difference between two opposite photocurrent flows generated from the top and bottom surfaces. Acting as an efficient SPGE material, Ag2Te demonstrates robust performance across a wide spectral range from visible to mid-infrared, making it promising for applications in solar cells and mid-infrared detectors. More importantly, SPGE generated on low-symmetric surfaces can potentially be found in various systems, thereby inspiring a broader range of choices for photovoltaic materials.

Supplementary Figure 1 | I-V      affected by tuned bandgap at different temperatures, especially the excitation taking place in the surface state.On the other hand, the two-terminal resistance of the device was also measured, which demonstrates a typical semiconductor behavior (Supplementary Figure 8d), opposite to the photocurrent.The drop in measured photocurrent is attributed to the effect of increased resistance, which influences the diffusion of the photocurrent.Similar temperature dependence is common in previous BPVE works 1,9 .We should emphasize that the photocurrent is generated at the surface of Ag2Te.However, the contribution of resistance is global (from surface, bulk, and contacts) because the whole device participates in propagating the photocurrent.

Supplementary Note 8: Theoretical calculation of shift current 8.1 Structural modeling:
To analyze the bulk photovoltaic effect (BPVE, or rather surface photogalvanic effect (SPGE)) of Ag2Te along the (1 ̅ 01) plane, we constructed a slab model from the Ag2Te crystal as shown in Supplementary Figure 9c, with both the top and bottom surfaces oriented along the (1 ̅ 01) plane.The process of slab extraction from the expanded unit cell is detailed in Supplementary Fig. 9, showing the alignment of the slab with the original crystal axes.A vacuum layer has been added along the c-axis of the slab model to eliminate interactions from periodic boundaries in this direction.We would use the slab model to represent an isolated surface structure in the following calculation.

Electronic structure calculations:
We performed electronic structure calculations using the Vienna Ab initio Simulation Package (VASP), adopting the Perdew-Burke-Ernzerhof (PBE) form of the generalized gradient approximation (GGA) for the exchange-correlation functional [10][11][12] .The kinetic energy cutoff for the plane-wave basis was set to 400 eV.For structural optimization and static self-consistent calculations, the Brillouin zone was sampled using Monkhorst-Pack k-point meshes of 9×17×1 and 15×27×1, respectively.The positions of the atoms were fully optimized with a force convergence criterion of -0.01eV/Å.To facilitate high-precision optical calculations, the energy convergence criterion for the electronic self-consistency was set to 10-7 eV.Additionally, a dipole correction was applied in the out-of-plane direction.

Calculation of nonlinear optical conductivity:
We utilize the Shift Current (SC) framework to address issues related to the BPVE (SPGE), which mainly describes the interband absorption.Considering the electric field component of a monochromatic light, it can be expressed as: where E represents the alternating current (AC) electric field of the light.
As a second-order optical response, the SC describes the direct current (DC) photocurrent output in the material's a-direction due to light polarized along the bdirection, expressed as: where    () represents the nonlinear optical conductivity induced by the SC, derived from Kubo formula.Note that    and the measured coefficient   in the main text can be converted with each other according to   = 2   /( 0 ), where  0 is the vacuum permittivity and  is the speed of light.   () can be expressed as 13 : It is crucial to note that, in calculations involving slab models, excluding the impact of the vacuum layer is essential for accurately determining the material's effective shift current output, formalized as: Where  represents the total thickness of the slab along the out-of-plane direction, and   denotes the material's effective layer thickness.Specifically, for our slab under discussion,  = 38.59Åand   = 13.26Å .For simplicity, the superscript 'eff' is omitted in subsequent diagrams and descriptions.
We implemented interpolation in k-space using the method of maximally localized Wannier functions (MLWFs), facilitated by the WANNIER90 code package 14,15 .This technique was specifically employed to improve the convergence of SC conductivity calculations in k-space.The interpolated k-point grid was expanded to 500×500×1.To specifically address the electronic properties near the Fermi level, trial orbitals for the Wannier projection were chosen as Ag-s and Te-p.The calculations of the aforementioned SC conductivity were conducted using the postw90-berry-sc module in the wannier90 post-processing program (postw90.x) 16.
The photon energy-dependent    () in the slab across different in-plane components are shown in Supplementary Figure 10a (with Cartesian directions  and  labeled in the figure).The non-zero terms are    and    , both maintaining consistent signs and trends within the energy windows.The tiny value in    is attributed to algorithmic errors.For comparison, similar calculations were performed for bulk Ag2Te (Supplementary Figure 10b), where all components consistently remain zero.
The calculation shows identical results with the qualitative symmetry analysis that a second-order nonlinear response along the b-axis exists on the surface of Ag2Te, while prohibited in the bulk.The resembled    and    also predict a unipolar and relatively weak polarization dependence of the measured photocurrent.Phenomenologically, the calculation captures the symmetry and polarization characteristics in the observed SPGE (Fig. 2 of the main text).
On the other hand, we didn't observe the calculated peak near 1.28eV and 1.89eV.There is probably no such practical laser wavelength (Fig. 4b of the main text) that completely coincides with the predicted energy.Moreover, the prominent contribution from specific transition is likely covered in the more complex actual band structure, especially within the large energy range.

Supplementary Figure 2 |
characteristic of the Ag2Te device shown in Fig. 1c of the main text.The scale bar is 5μm.Bias current dependence of the photovoltage response in Ag2Te.a,b, Optical image (a) and reflection image (b) of the Ag2Te device.c, Lineprofile of the photovoltage with different bias currents extracted from d along the black arrows.The two orange-pink regions indicate the electrodes.d, Photovoltage mapping with different bias currents.The laser wavelength is 690nm and the power is ~2.4uW.The scale bars are all 5μm.

Supplementary Figure 8 |
photocurrent (nA) Photocurrent (nA) denotes the spin degeneracy,   =   −   and ℏ  =   −   are the differences in occupation numbers and energy eigenvalues between bands indexed by |∂    ⟩ denotes the cell-periodic part of the Bloch eigenstate, and ∂  is shorthand for /   .